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hi there

i just started an algebra class and my teacher gave us this problem to do for homework and I need some help on working though it:

 

Find all solutions to the system \begin{align*} a + b &= 14, \\ a^3 + b^3 &= 812. \end{align*}

 

tysm in advance!! :D

 Feb 15, 2022
 #1
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from equation 1 we get b = 14-a

 

substituting that into the second equation we get a^3+(14-a)^3=812

 

now you can try to expand it out, if you need help just ask

 Feb 15, 2022
 #3
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You will end up with more complicated polynomials, and the person who asked the question may not be able to fully expand it out or solve it with a cubic equation.

 

Though if you do factor it out it may be a better strategy if you know how... :)

proyaop  Feb 15, 2022
 #2
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So a^3 + b^3 = (a + b)(a^2 - ab + b^2). 

Substituting in, we have 812 = 14(a^2 - ab + b^2).

Simplifying we have 58 = a^2 - ab + b^2.

 

a^2 - ab + b^2 = (a + b)(a + b) - 3ab. 

Substituting in, we have 58 = (14)(14) - 3ab.

Now we have ab = 46.

 

Then we also have a^2 - ab + b^2 = (a - b)(a - b) + ab.

Thus, 58 = (a - b)^2 + 46.

(a - b)^2 = 12.

\(a - b = \pm{2\sqrt{3}}\)

 

Now we can sove for a. If we add the two equations together, we get 2a = 14 + 2sqrt(3)

Thus, \(a = 7 + \sqrt{3}\), and \(a = 7 - \sqrt{3}\).

Substituting in, we have \(b = 7 - \sqrt{3}\), and \(b = 7 - 3\sqrt{3}\).

 

Thus, our possible for solutions \((a, b)\)are:

\((7 + \sqrt{3}, 7 - \sqrt{3})\)

\((7 - \sqrt{3}, 7 - 3\sqrt{3})\)

 

smiley

 Feb 15, 2022
 #4
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omg thank you so much!!

 Feb 16, 2022
 #5
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No problem, just remember the fact that whenever you encounter a hard system of equations problem, either use substitution or try factoring the shenanigans before brute forcing the question with guess and check.

 

Work smart not hard... :D yw

proyaop  Feb 16, 2022

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